Kobon Triangles

Kobon Fujimura, a Japanese puzzle expert, recently invented a problem in combinatorial geometry. It is simple to state, but no general solution has yet been found. What is the largest number of nonoverlapping triangles that can be produced by n straight line segments?

It is not hard to discover by trial and error that for n = 3, 4, 5 and 6 the maximum number of triangles is respectively one, two, five and seven [see Figure 108]. For seven lines the problem is no longer easy. The reader is asked to search for the maximum number of nonoverlapping triangles that can be produced by seven, eight and nine lines.

The problem of finding a formula for the maximum 'number of triangles as a function of the number of lines appears to be extremely difficult.

Figure 108

Figure 108

Maximum number of nonoverlapping triangles for three, four, five, and six lines

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